End of Semester Report
Spring 2005
Jeremy Morris
May 2, 2005
At the beginning of the semester I proposed to do two things. The
first was to alter the model I was using to determine relationships between
students at the university. The second was to create a web site where I was
going to document my results and progress on this project.
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A More Realistic Model
The main aim of this project has been to see if we can establish whether or
not the University of Utah meets the criteria for a small world, as defined
by Duncan Watts. To do this, I have received some data from the university
where we can create a graph of students where we place an edge in the graph
between students that take at least on class together. We have shown that
this graph can be considered a small world. However, if we think about
what really happens in the classroom, it is not realistic to think that any
given student knows or interacts with, all other students in each course. So
we attempted to create a more realistic graph.
In this more realistic graph, we connect students with probability
p
kU
ni
(1)
Where p is a parameter chosen by us, ni is the number of students attending
the ith course and kU is the average class size in our data set (this number
is approximately 20). What this means is that in an average sized class,
a student will know a certain percentage of the people in that class. This
percentage is scaled with the size of the class such that in a larger class,
the student knows less people, and in a smaller class, the student will know
more. Experience tells us that this closely approximates reality.
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We ran a few experiments to see what value of p would give us a realistic
number of average connections. It turns out that if we pick p = 0.40, we
get an average degree of 17, this is down from 208 if we connect students to
everyone in all their courses. For this graph we have the results
Graph Statistics
Number of Students in Graph
Avg Degree
Characteristic Path Length (L)
Clustering Coefficient (γ)
28517
17
4.52656
0.182163
This shows that, amazingly, the U can be considered a small world even
when we make drastic restrictions on how we connect the graph. It is also
amazing that the characteristic path length (which measures the typical
distance between nodes) only increased slightly from 2.88 to 4.52. This is a
very short distance for a disease to travel.
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Documentation and the Web site
For a large part of the semester I have been working on a web site that will
document this project. The web site will eventually feature an explanation
of the project, results from the various experiments I have run, the software
I have written and an interactive section for students. Some of these things
are not ready for release and will be added sometime this summer.
The interactive section will allows students to enter their course schedule
so that they can tell a few things about themselves. After entering a course
schedule, students will be able to see their Google weight, their mean path
length and their clustering coefficient. There are links to definitions for all
of these statistics.
The web site can be found at
http://sundial.math.utah.edu/smallU/
I have had a very good experience with the REU program and thank
everyone involved for giving me the opportunity to participate. Thank you.
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